In the first session we introduce the fundamental concepts of multigrid computational methods, followed by the classical geometric multigrid algorithm and ending with analytical tools for algorithm development and assessment based on Fourier analysis. We begin with a display of the basic observations that generally underlie multilevel algorithms and identify four essential building blocks that need to be selected. We then show how these ingredients come together in the classical geometric multigrid algorithm for the numerical solution of linear elliptic partial differential equations on rectangular grids. Finally, we introduce a Fourier analysis tool known as Local Mode Analysis (LMA), which allows us to estimate quantitative approximate convergence rates for the multigrid algorithm, and thus to predict its behavior, to debug code, and to skillfully tune parameters.

In the practice session we put theory to the test. Beginning with a basic multigrid solver for the Poisson problem on a square grid (which will be provided), we test its convergence properties in practice, extend it to more general problems, and compare its performance against theoretical predictions based on LMA.

The material in this introduction is summarized in part in the informal paper :

"MG_Damped_Jacobi.m" is a MATLAB program that solves the Two-Dimensional Poisson problem on a square using multigrid iterations (V-Cycles), and plots (1) The residual norm ; (2) The residual norm reduction factor per iteration ; (3) The final numerical solution.

In the practice session we will modify and develop this program. To make the best of the practice session, it is recommended to examine this code in advance and to make sure it works properly on your computer. To run, enter

MG_Damped_Jacobi(n)

where n is some power of 2, e.g., n = 128. This will solve the problem on a grid of n by n mesh intervals. The code should also run on the freely available packages, such as FreeMat.

Multigrid Methods provide optimal properties as an iterative solution method for a large class of linear systems. Multigrid methods were originally developed to solve boundary value problems on spatial domains. By choosing a discretization method, such problems become systems of algebraic equations associated with the spatial discretization. Then, in multigrid, the solution to the problem at hand is reconstructed using information from some coarser representation of the problem. It is made use of the fact that simple and cheap iterative smoothing processes are efficient in reducing high-oscillatory error components whereas are not able to address low-frequency errors. By using different resolutions of the problem through multiple grids, one can effectively address all frequencies of the solution.

In contrary to geometric multigrid, where explicit hierarchies of spatial discretizations are utilized, an algebraic multigrid approach is not based on meshes but uses purely algebraic definitions of coarse representations of the fine grid problem. This makes algebraic multigrid an ideal approach to be used on complicated geometries and unstructured meshes, where explicit coarse discretizations are tedious to construct.

In this presentation, the focus will be on so called smoothed aggregation algebraic multigrid methods (SA-AMG). After introducing the principle multigrid idea, notation and framework, the SA-AMG formulation will be discussed with special focus on parallel implementations issues and applicability to a range of problems such as computational structural and fluid dynamics problems. The hands on tutorial allows participants to get acquainted with the open source parallel AMG code MueLu, the choice of its parameters and its behavior by applying it to predefined sample problems.

10h30-11h :Pause

11h-12h30 :An Introduction to Algebraic Multigrid Methods Based on Aggregation

In this lecture, we present algebraic multigrid (AMG) methods based on plain (or unsmoothed) aggregation of the unknowns. We start with some theoretical and practical motivation, and next we discuss how to properly choose the multigrid components in this context (the smoother, the number of smoothing steps, the multigrid cycle). Doing so, we highlight the main differences with respect to geometric multigrid and classical AMG. We finally present the key ideas for an efficient parallelization.

Linear systems that arise in industrial simulations tend to be smaller
in size than their counterparts in research, but not necessarily easier
to solve. This presentation will review different multilevel methods
that have been evaluated at EDF over the last decade. For certain
differential operators, they have become the default choice as linear
solver. However, for some other applications, the design of a fast and
robust method remains an open question — for now.